Vol. 8, No. 10, 2014

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Explicit points on the Legendre curve III

Douglas Ulmer

Vol. 8 (2014), No. 10, 2471–2522
Abstract

We continue our study of the Legendre elliptic curve ${y}^{2}=x\left(x+1\right)\left(x+t\right)$ over function fields ${K}_{d}={\mathbb{F}}_{p}\left({\mu }_{d},{t}^{1∕d}\right)$. When $d={p}^{f}+1$, we have previously exhibited explicit points generating a subgroup ${V}_{d}\subset E\left({K}_{d}\right)$ of rank $d-2$ and of finite, $p$-power index. We also proved the finiteness of Ш$\left(E∕{K}_{d}\right)$ and a class number formula: ${\left[E\left({K}_{d}\right):{V}_{d}\right]}^{2}=|Ш\left(E∕{K}_{d}\right)|$. In this paper, we compute $E\left({K}_{d}\right)∕{V}_{d}$ and Ш$\left(E∕{K}_{d}\right)$ explicitly as modules over ${ℤ}_{p}\left[Gal\left({K}_{d}∕{\mathbb{F}}_{p}\left(t\right)\right)\right]$.

Keywords
elliptic curves, function fields, Tate–Shafarevich group
Mathematical Subject Classification 2010
Primary: 11G05, 14G05
Secondary: 11G40, 14K15